List of Contributors Contributors Preface 1 Introduction Clair Alston, Margaret Donald, Kerrie Mengersen and Anthony Pettitt 1.1 Introduction 1.2 Overview 1.3 Further Reading 1.3.1 Bayesian theory and methodology 1.3.2 Bayesian Theory and Methodology 1.3.3 Bayesian Computation 1.3.4 Bayesian Software 1.3.5 Applications References 2 Introduction to MCMC Anthony N. Pettitt and Candice M. Hincksman 2.1 Introduction 2.2 Gibbs Sampling 2.2.1 Example: Bivariate normal. 2.2.2 Example: Change point model 2.3 Metropolis-Hastings algorithms 2.3.1 Example: Component wise MH or MH within Gibbs 2.3.2 Extensions to basic MCMC 2.3.3 Adaptive MCMC 2.3.4 Doubly intractable problem 2.4 Approximate Bayesian Computation (ABC) 2.5 Reversible Jump Markov chain Monte Carlo 2.6 MCMC for some further applications References 3Priors: Silent or Active Partners of Bayesian Inference? Samantha Low-Choy 3.1 Priors in the very beginning 3.1.1 Priors as a basis for Learning 3.1.2 Priors and Philosophy 3.1.3 Prior chronology 3.1.4 Pooling Prior Information 3.2 Methodology I: Priors defined by mathematical criteria 3.2.1 Conjugate Priors 3.2.2 Conjugacy for a Normal prior on the mean, in a Normal likelihood. 3.2.3 Conjugacy for a Beta prior on the probability of success, with a Binomial likelihood; see Gelman et al. (2004). 3.2.4 Conjugate prior for Normal linear regression; see Gelman et al. (2004). 3.2.5 Conditionally conjugate priors for random effects variances. 3.2.6 Impropreity and Hierarchical Priors 3.2.7 Zellner's g -prior for regression models. 3.2.8 Objective priors 3.3 Methodology II: Modelling Informative Priors 3.3.1 Informative modelling approaches 3.3.2 Elicitation of distributions 3.4 Case studies 3.4.1 Normal likelihood: Time to submit research dissertations 3.4.2 Binomial likelihood: Surveillance for exotic plant pests 3.4.3 Mixture model likelihood: Bioregionalisation 3.4.4 Logistic regression likelihood: Mapping species distribution via habitat models 3.5 Discussion 3.5.1 Limitations. 3.5.2 Finding out about the problem. 3.5.3 Prior formulation. 3.5.4 Communication. 3.5.5 Conclusion 3.6 Acknowledgements References 4 Bayesian analysis of the Normal linear regression model Christopher M. Strickland and Clair. L. Alston 4.1 Introduction 4.2 Case Studies 1 4.2.1 Case Study 1: Boston Housing Data Set 4.2.2 Case Study 2: Production of Cars and Station wagons 4.3 Matrix notation and the likelihood 4.4 Posterior Inference 4.4.1 Natural Conjugate Prior 4.4.2 Alternative Prior Specifications 4.4.3 Generalisations of the normal linear model 4.4.4 Variable Selection 4.5 Analysis 4.5.1 Case Study : Boston housing data set 4.5.2 Case Study 2: Car production data set References 5 Adapting ICU mortality models for local data: A Bayesian approach Petra L. Graham, Kerrie L. Mengersen and David A. Cook 5.1 Introduction 5.2 Case study: Updating a known risk-adjustment model for local use 5.3 Models and Methods 5.4 Data analysis and Results 5.4.1 Updating using the training data 5.4.2 Updating the model yearly 5.5 Discussion References 6 A Bayesian Regression Model with Variable Selection for Genome-Wide Association Studies Carla Chen, Kerrie L. Mengersen, Katja Ickstadt and Jonathan M. Keith 6.1 Introduction 6.2 Case study: Case-Control of Type I diabetes 6.3 Case study: GENICA 6.4 Models and Methods 6.4.1 Main effect models 6.4.2 Main effects and interactions 6.5 Data Analysis and Results 6.5.1 WTCCC-Type I diabetes 6.5.2 Genica 6.6 Discussion References 6.A SNP IDs 7 Bayesian Meta-Analysis Jegar O. Pitchforth and Kerrie L. Mengersen 7.1 Introduction 7.2 Case Study 1: association between red meat consumption and breast cancer 7.2.1 Background 7.2.2 Meta-analysis models 7.2.3 Computation 7.2.4 Results 7.2.5 Discussion 7.3 Case study 2: Trends in fish growth rate and size 7.3.1 Background 7.3.2 Meta-analysis models 7.3.3 Computation 7.3.4 Re